Adjoints and canonical forms of tree amplituhedra
DOI:
https://doi.org/10.7146/math.scand.a-149816Abstract
We consider semi-algebraic subsets of the Grassmannian of lines in three-space called tree amplituhedra. These arise in the study of scattering amplitudes from particle physics. Our main result states that tree amplituhedra in Gr(2,4) are positive geometries. The numerator of their canonical form plays the role of the adjoint in Wachspress geometry, and is uniquely determined by explicit interpolation conditions.
References
Arkani-Hamed, N., Bai, Y., and Lam, T., Positive geometries and canonical forms, J. High Energy Phys. 2017, no. 11, 039, front matter+121 pp. https://doi.org/10.1007/jhep11(2017)039
Arkani-Hamed, N., Hodges, A., and Trnka, J., Positive amplitudes in the amplituhedron, J. High Energy Phys. 2015, no. 8, 030, front matter+24 pp. https://doi.org/10.1007/JHEP08(2015)030
Arkani-Hamed, N., Thomas, H., and Trnka, J., Unwinding the amplituhedron in binary, J. High Energy Phys. 2018, no. 1, 016, front matter+40 pp. https://doi.org/10.1007/jhep01(2018)01
Arkani-Hamed, N., and Trnka, J., The amplituhedron, Journal of High Energy Physics 2014, no. 10, 030, front matter+30 pp. https://doi.org/10.1007/JHEP10(2014)030
Even-Zohar, C., Lakrec, T., Parisi, M., Tessler, R., Sherman-Bennett, M., and Williams, L., A cluster of results on amplituhedron tiles, Lett. Math. Phys. 114 (2024), no. 5, Paper No. 111. https://doi.org/10.1007/s11005-024-01854-4
Franco, S., Galloni, D., Mariotti, A., and Trnka, J., Anatomy of the amplituhedron, J. High Energy Phys. 2015, no. 3, 128, front matter+61 pp. https://doi.org/10.1007/JHEP03(2015)128
Fulton, W., Intersection theory, Second edition. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. Springer-Verlag, Berlin, 1998. https://doi.org/10.1007/978-1-4612-1700-8
Gaetz, C., Positive geometries learning seminar, unpublished lecture notes, available at https://math.mit.edu/ tfylam/posgeom/gaetz_notes.pdf (2020).
Galashin, P., Karp, S. N., and Lam, T., The totally nonnegative Grassmannian is a ball, Adv. Math. 397 (2022), Paper No. 108123, 23 pp. https://doi.org/10.1016/j.aim.2021.108123
Kohn, K., Piene, R., Ranestad, K., Rydell, F., Shapiro, B., Sinn, R., Sorea, M.-S., and Telen, S., Adjoints and canonical forms of polypols, arXiv:2108.11747 (2021).
Kohn, K., and Ranestad, K., Projective geometry of Wachspress coordinates, Found. Comput. Math. 20 (2020), no. 5, 1135–1173. https://doi.org/10.1007/s10208-019-09441-z
Lam, T., An invitation to positive geometries, arXiv:2208.05407.
Łukowski, T., On the boundaries of the m=2 amplituhedron, Ann. Inst. Henri Poincaré D 9 (2022), no. 3, 525–541. https://doi.org/10.4171/aihpd/124
Łukowski, T., Parisi, M., and Williams, L. K., The positive tropical Grassmannian, the hypersimplex, and the m=2 amplituhedron, Int. Math. Res. Not. IMRN 2023, no. 19, 16778–16836. https://doi.org/10.1093/imrn/rnad010
Mandelshtam, Y., Pavlov, D., and Pratt, E., Combinatorics of m=1 grasstopes, arXiv:2307.09603.
Parisi, M., Sherman-Bennett, M., and Williams, L., The m=2 amplituhedron and the hypersimplex: Signs, clusters, tilings, Eulerian numbers, Commun. Am. Math. Soc. 3 (2023), 329–399. https://doi.org/10.1090/cams/23
Sinn, R., Algebraic boundaries of (mathrm SO(2))-orbitopes, Discrete Comput. Geom. 50 (2013), no. 1, 219–235. https://doi.org/10.1007/s00454-013-9501-5
Sturmfels, B., Totally positive matrices and cyclic polytopes, Linear Algebra Appl. 107 (1988), 275–281. https://doi.org/https://doi.org/10.1016/0024-3795(88)90250-9
